Quite a popular request on my job sheets ...' Re-do silicone around sink and bath'...
Every time I tackle them in the same way. A stanley knife is used to cut the old silicone out. Cut along the back edge of the old silicone, sliciing right down to the sill. Then cut along the bottom edge slicing right back to the window frame.
If you are left with quite a bit of the old sicone still sticking to either surface, use a scraper to remove as much as you can. Personally I would continue to scrape until all the old stuff is removed. Anything left can be got rid of with a siicone remover if need be, but that just takes more time. ( and we all know what more time costs).
As for the new siiicone, the best one I have found is called 'Forever White' .. it is made by Everbuild and seems to have no issues with mould or yellowing.
When it first came out I was paying £15 a tube for it, but it was worth it.
Now it can be got at a more realistic price ...
http://www.toolstation.com/shop/p63890?searchstr=forever%20white
Just over fill the gap and finish it of with a fugi ..
http://www.toolstation.com/shop/p46602?searchstr=fug
.. people will think you are a real pro when you use one of these.